124 Fundamental Concept of Reliability Based Structural Design

It is assumed in the following sections that the readers are familiar with the basic concept of uncertainty analysis. If not, they are urged to refer to a recent book authored by Haldar and Mahadevan [14].

In general, R and S can be used to represent the resistance and load effect as random variables since they are functions of many other random variables. R is a function of material properties and the geometric properties of a structural element including cross-sectional properties. S is a function of the load effect that can be expected during the lifetime of the structural element. The uncertainty in R and S can be completely defined by their corresponding probability density functions (PDFs) denoted as fR(r) and fS(s), respectively. Then, the probability of failure of the structural element can be defined as the probability of the resistance being less than the load effect or simply P(R > S). Mathematically, it can be expressed as [14]

P(failure)= P(R < S)= fR(r) dr fs(s) ds = Fr(sf(s) ds (12.2)

Jo Uo J Jo where FR(s) is the cumulative distribution function (CDF) of R evaluated at s. Conceptually, Equation 12.2 states that for a particular value of the random variable S = s, FR(s) is the probability of failure. However, since S is also a random variable, the integration needs to be carried out for all possible values of S, with their respective likelihood represented by the corresponding PDF. Equation 12.2 can be considered as the fundamental equation of the reliability-based design concept.